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<h1 class="heading"><a href="MATH-2023-OPDE.html"><span class="title">MATH 2023: Ordinary and Partial Differential Equations</span></a></h1>
<p class="byline">Xiaoyi Chen and Wei Zhang</p>
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<a href="ch_first.html" data-scroll="ch_first" class="internal"><span class="codenumber">1</span> <span class="title">Introduction</span></a><ul>
<li><a href="sec_1-intro.html" data-scroll="sec_1-intro" class="internal">Classification of Differential Equations</a></li>
<li><a href="sec_2-intro.html" data-scroll="sec_2-intro" class="internal">Linear and Nonlinear Equation</a></li>
<li><a href="sec_3-intro.html" data-scroll="sec_3-intro" class="internal">Geometrical Aspect</a></li>
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<a href="ch_second.html" data-scroll="ch_second" class="internal"><span class="codenumber">2</span> <span class="title">First Order Ordinary Differential Equations</span></a><ul>
<li><a href="sec2_1.html" data-scroll="sec2_1" class="internal">Linear Equations</a></li>
<li><a href="sec2_2.html" data-scroll="sec2_2" class="internal">Further Discussion of Linear Equations (For reading only)</a></li>
<li><a href="sec2_3.html" data-scroll="sec2_3" class="internal">Separable Equations</a></li>
<li><a href="sec2_4.html" data-scroll="sec2_4" class="internal">Difference Between Linear and Nonlinear Equations</a></li>
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<li><a href="sec2_6.html" data-scroll="sec2_6" class="internal">Exact Equations and Integrating Factors</a></li>
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<a href="ch_third.html" data-scroll="ch_third" class="internal"><span class="codenumber">3</span> <span class="title">third Order Linear Equations</span></a><ul>
<li><a href="sec3_1.html" data-scroll="sec3_1" class="internal">Homogeneous equations with constant coefficient</a></li>
<li><a href="sec3_2.html" data-scroll="sec3_2" class="internal">Fundamental Solutions of Linear Homogeneous Equations</a></li>
<li><a href="sec3_3.html" data-scroll="sec3_3" class="internal">Linear Independence and Wronskian</a></li>
<li><a href="sec3_4.html" data-scroll="sec3_4" class="internal">Complex roots of the characteristic equations</a></li>
<li><a href="sec3_5.html" data-scroll="sec3_5" class="internal">Repeated Roots: Reduction of Order</a></li>
<li><a href="sec3_6.html" data-scroll="sec3_6" class="internal">Non-homogeneous Equations and Method of Undetermined Coefficients</a></li>
<li><a href="sec3_7.html" data-scroll="sec3_7" class="internal">Variation of Parameters</a></li>
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<a href="ch_four.html" data-scroll="ch_four" class="internal"><span class="codenumber">4</span> <span class="title">Higher Order Linear Equations</span></a><ul>
<li><a href="sec4_1.html" data-scroll="sec4_1" class="active">General Theory of the <span class="process-math">\(n\)</span>-th Order Linear Equations</a></li>
<li><a href="sec4_2.html" data-scroll="sec4_2" class="internal">Homogeneous Equations with Constant Coefficients</a></li>
<li><a href="sec4_3.html" data-scroll="sec4_3" class="internal">The Method of Undetermined Coefficients</a></li>
<li><a href="sec4_4.html" data-scroll="sec4_4" class="internal">The Method of Variation of Parameters</a></li>
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<a href="ch_five.html" data-scroll="ch_five" class="internal"><span class="codenumber">5</span> <span class="title">Series Solutions of Second Order Linear Equations</span></a><ul>
<li><a href="sec5_1.html" data-scroll="sec5_1" class="internal">Brief Review on Power Series</a></li>
<li><a href="sec5_2.html" data-scroll="sec5_2" class="internal">Introduction</a></li>
<li><a href="sec5_3.html" data-scroll="sec5_3" class="internal">Series Solutions Near an Ordinary Point</a></li>
<li><a href="sec5_4.html" data-scroll="sec5_4" class="internal">Euler’s Equation</a></li>
<li><a href="sec5_5.html" data-scroll="sec5_5" class="internal">Series Solution near a Regular Singular Point</a></li>
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<a href="ch_six.html" data-scroll="ch_six" class="internal"><span class="codenumber">6</span> <span class="title">System of First Order Linear Equations</span></a><ul>
<li><a href="sec6_1.html" data-scroll="sec6_1" class="internal">Introduction <span class="process-math">\(\&amp;\)</span> Basic Theory</a></li>
<li><a href="sec6_2.html" data-scroll="sec6_2" class="internal">Homogeneous System with Constant Coefficients</a></li>
<li><a href="sec6_3.html" data-scroll="sec6_3" class="internal">Complex Eigenvalues</a></li>
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<li><a href="sec6_5.html" data-scroll="sec6_5" class="internal">Fundamental Matrices</a></li>
<li><a href="sec6_6.html" data-scroll="sec6_6" class="internal">Non-homogeneous linear systems</a></li>
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<a href="ch_seven.html" data-scroll="ch_seven" class="internal"><span class="codenumber">7</span> <span class="title">Partial Differential Equations</span></a><ul>
<li><a href="sec7_1.html" data-scroll="sec7_1" class="internal">Two-Point Boundary Value Problems</a></li>
<li><a href="sec7_2.html" data-scroll="sec7_2" class="internal">Eigenvalue Problems</a></li>
<li><a href="sec7_3.html" data-scroll="sec7_3" class="internal">Fourier Series</a></li>
<li><a href="sec7_4.html" data-scroll="sec7_4" class="internal">The Fourier Convergence Theorem</a></li>
<li><a href="sec7_5.html" data-scroll="sec7_5" class="internal">Even and Odd Functions</a></li>
<li><a href="sec7_6.html" data-scroll="sec7_6" class="internal">Introduction to Partial Differential Equations</a></li>
<li><a href="sec7_7.html" data-scroll="sec7_7" class="internal">1D Heat Equation; Solutions by Separation of Variable and Fourier Series</a></li>
<li><a href="sec7_8.html" data-scroll="sec7_8" class="internal">Other Heat Conduction Problems</a></li>
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<a href="ch_eight.html" data-scroll="ch_eight" class="internal"><span class="codenumber">8</span> <span class="title">Laplace transform</span></a><ul>
<li><a href="sec8_1.html" data-scroll="sec8_1" class="internal">What are Laplace Transforms, and Why?</a></li>
<li><a href="sec8_2.html" data-scroll="sec8_2" class="internal">Finding Laplace Transforms</a></li>
<li><a href="sec8_3.html" data-scroll="sec8_3" class="internal">Finding inverse transforms using partial fractions</a></li>
<li><a href="sec8_4.html" data-scroll="sec8_4" class="internal">Solving ODEs and ODE Systems</a></li>
<li><a href="sec8_5.html" data-scroll="sec8_5" class="internal">Step input and Impulse problems</a></li>
<li><a href="sec8_6.html" data-scroll="sec8_6" class="internal">Laplace transform for PDE (heat equation)</a></li>
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<li class="link"><a href="solutions-1.html" data-scroll="solutions-1" class="internal"><span class="codenumber">A</span> <span class="title">Selected Hints</span></a></li>
<li class="link"><a href="solutions-2.html" data-scroll="solutions-2" class="internal"><span class="codenumber">B</span> <span class="title">Selected Solutions</span></a></li>
<li class="link"><a href="appendix-1.html" data-scroll="appendix-1" class="internal"><span class="codenumber">C</span> <span class="title">List of Symbols</span></a></li>
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<main class="main"><div id="content" class="pretext-content"><section class="section" id="sec4_1"><h2 class="heading hide-type">
<span class="type">Section</span> <span class="codenumber">4.1</span> <span class="title">General Theory of the <span class="process-math">\(n\)</span>-th Order Linear Equations</span>
</h2>
<p id="p-145">All the results for the second order linear equations can be naturally extended to the <span class="process-math">\(n\)</span>-th order linear equations. Consider</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/eq4_1.html" id="eq4_1">
\begin{equation}
\frac{\textrm{d}^n y}{\textrm{d} x^n}+P_1(x) \frac{\textrm{d}^{n-1} y}{\textrm{d} x^{n-1}}+\cdots+P_{n-1}(x) \frac{\textrm{d} y}{\textrm{d} x}+P_n(x) y=g(x).\tag{4.1.1}
\end{equation}
</div>
<p class="continuation">Introduce</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/eq4_1.html">
\begin{equation*}
L[y]=\frac{\textrm{d}^n y}{\textrm{d} x^n}+P_1(x) \frac{\textrm{d}^{n-1} y}{\textrm{d} x^{n-1}}+\cdots+P_{n-1}(x) \frac{\textrm{d} y}{\textrm{d} x}+P_n(x) y.
\end{equation*}
</div>
<p class="continuation">Then (<a href="" class="xref" data-knowl="./knowl/eq4_1.html" title="Equation 4.1.1">(4.1.1)</a>) can be rewritten as</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/eq4_1.html" id="eq4_1_1">
\begin{equation}
L[y]=g(x).\tag{4.1.2}
\end{equation}
</div>
<p class="continuation">Initial conditions are proposed as</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/eq4_1.html" id="eq4_2">
\begin{equation}
y(x_0)=b_0,\quad y^{\prime}(x_0)=b_1,\cdots, y^{(n-1)}(x_0)=b_{n-1}.\tag{4.1.3}
\end{equation}
</div>
<p id="p-146"><dfn class="terminology">Theorem</dfn> If <span class="process-math">\(P_1(x), P_2(x),\cdots, P_n(x), g(x)\)</span> are continuous on an open interval <span class="process-math">\(I\text{,}\)</span> then the initial value problem (<a href="" class="xref" data-knowl="./knowl/eq4_1_1.html" title="Equation 4.1.2">(4.1.2)</a>) and (<a href="" class="xref" data-knowl="./knowl/eq4_2.html" title="Equation 4.1.3">(4.1.3)</a>) has a unique solution which is valid in <span class="process-math">\(I\text{.}\)</span></p>
<p id="p-147"><dfn class="terminology">The Homogeneous Equation</dfn></p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
L[y]=0.
\end{equation*}
</div>
<p class="continuation">Suppose that <span class="process-math">\(y_1, y_2,\cdots, y_n\)</span> are <span class="process-math">\(n\)</span> solutions. Then</p>
<div class="displaymath process-math" data-contains-math-knowls="" id="eq4_3">
\begin{equation}
y=C_1 y_1+C_2 y_2+\cdots+C_n y_n\tag{4.1.4}
\end{equation}
</div>
<p class="continuation">is also a solution.</p>
<p id="p-148">Is (<a href="" class="xref" data-knowl="./knowl/eq4_3.html" title="Equation 4.1.4">(4.1.4)</a>) the general solution? To answer this, we need to see whether <span class="process-math">\(C_1, C_2, \cdots, C_n\)</span> can be uniquely determined under any initial conditions.</p>
<div class="displaymath process-math" data-contains-math-knowls="" id="p-149">
\begin{equation*}
\begin{aligned}
&amp; y(x_0)=b_0 \to C_1 y_1(x_0)+C_2 y_2(x_0)+\cdots+C_n y_n(x_0)=b_0,\\
&amp;y^{\prime}(x_0)=b_1 \to C_1 y_1^{\prime}(x_0)+C_2 y_2^{\prime}(x_0)+\cdots+C_n y_n^{\prime}(x_0)=b_1,\\
&amp;\cdots\\
&amp;y^{(n-1)}(x_0)=b_{n-1} \to C_1 y_1^{(n-1)}(x_0)+C_2 y_2^{(n-1)}(x_0)+\cdots+C_n y_n^{(n-1)}(x_0)=b_{n-1}.
\end{aligned}
\end{equation*}
</div>
<p class="continuation">These are the <span class="process-math">\(n\)</span> equations for <span class="process-math">\(C_1, C_2,\cdots, C_n\text{.}\)</span> These equations have unique solution to <span class="process-math">\(C_1, C_2, \cdots, C_n\)</span> if and only if</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
W_0=\left|
\begin{array}{cccc}
y_1(x_0) &amp; y_2(x_0) &amp; \cdots &amp; y_n(x_0)\\
y_1^{\prime}(x_0) &amp; y_2^{\prime}(x_0) &amp; \cdots &amp; y_n^{\prime}(x_0)\\
&amp; &amp; \cdots &amp; \\
y_1^{(n-1)}(x_0) &amp; y_2^{(n-1)} (x_0) &amp; \cdots &amp; y_n^{(n-1)}(x_0)
\end{array}
\right|
\neq 0.
\end{equation*}
</div>
<p id="p-150">We define the Wronskian for <span class="process-math">\(y_1, y_2, \cdots, y_n\)</span> as</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
W( y_1, \cdots, y_n)=\left|
\begin{array}{cccc}
y_1(x) &amp; y_2(x) &amp; \cdots &amp; y_n(x)\\
y_1^{\prime}(x) &amp; y_2^{\prime}(x) &amp; \cdots &amp; y_n^{\prime}(x)\\
&amp; &amp; \cdots &amp; \\
y_1^{(n-1)}(x) &amp; y_2^{(n-1)} (x) &amp; \cdots &amp; y_n^{(n-1)}(x)
\end{array}
\right|.
\end{equation*}
</div>
<p class="continuation">Then</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
W_0=W|_{x=x_0}.
\end{equation*}
</div>
<p id="p-151"><dfn class="terminology">Theorem</dfn> Suppose that <span class="process-math">\(P_1(x), P_2(x), \cdots, P_n(x)\)</span> are continuous on an open interval <span class="process-math">\(I\text{.}\)</span> If <span class="process-math">\(W(y_1, y_2, \cdots, y_n)\)</span> is not equal to zero at a point <span class="process-math">\(x_0\)</span> in <span class="process-math">\(I\)</span> (it can be shown that this implies <span class="process-math">\(W(y_1, y_2, \cdots, y_n) \neq 0\)</span> on the whole interval). Then (<a href="" class="xref" data-knowl="./knowl/eq4_3.html" title="Equation 4.1.4">(4.1.4)</a>) is the general solution and <span class="process-math">\(y_1, y_2, \cdots, y_n\)</span> are then called a fundamental set of solutions.</p>
<p id="p-152">Consider <span class="process-math">\(f_1(x), f_2(x), \cdots, f_n(x)\text{.}\)</span> If</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
k_1 f_1+k_2 f_2+\cdots k_n f_n=0,
\end{equation*}
</div>
<p class="continuation">implies <span class="process-math">\(k_1=k_2=\cdots=k_n=0\text{,}\)</span> then <span class="process-math">\(f_1, f_2, \cdots, f_n\)</span> are linear independent. Otherwise, they are linear dependent.</p>
<p id="p-153"><dfn class="terminology">Necessary and Sufficient Conditions</dfn> A set of solutions to <span class="process-math">\(L[y]=0\text{,}\)</span> <span class="process-math">\(\{y_1\text{,}\)</span> <span class="process-math">\(y_2\text{,}\)</span><span class="process-math">\(\cdots\text{,}\)</span> <span class="process-math">\(y_n\}\text{,}\)</span> are linear independent if and only if <span class="process-math">\(W(y_1, y_2, \cdots, y_n)\neq 0\)</span> at a point.</p>
<p id="p-154"><dfn class="terminology">The Nonhomogeneous Equations</dfn></p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/eq4_4.html ./knowl/eq4_5.html ./knowl/eq4_4.html ./knowl/eq4_4.html" id="eq4_4">
\begin{equation}
L[y]=g(x).\tag{4.1.5}
\end{equation}
</div>
<p class="continuation">If <span class="process-math">\(Y_1\)</span> and <span class="process-math">\(Y_2\)</span> are two solutions of (<a href="" class="xref" data-knowl="./knowl/eq4_4.html" title="Equation 4.1.5">(4.1.5)</a>). Then <span class="process-math">\(Y_1-Y_2\)</span> is a solution of</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/eq4_4.html ./knowl/eq4_5.html ./knowl/eq4_4.html ./knowl/eq4_4.html" id="eq4_5">
\begin{equation}
L[y]=0.\tag{4.1.6}
\end{equation}
</div>
<p class="continuation">If <span class="process-math">\(y_1, y_2, \cdots, y_n\)</span> are a fundamental set of solutions of (<a href="" class="xref" data-knowl="./knowl/eq4_5.html" title="Equation 4.1.6">(4.1.6)</a>), and <span class="process-math">\(Y(x)\)</span> is a particular solution of (<a href="" class="xref" data-knowl="./knowl/eq4_4.html" title="Equation 4.1.5">(4.1.5)</a>), then</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/eq4_4.html ./knowl/eq4_5.html ./knowl/eq4_4.html ./knowl/eq4_4.html">
\begin{equation*}
y=C_1 y_1+C_2 y_2+\cdots+C_n y_n+Y(x)
\end{equation*}
</div>
<p class="continuation">is the general solution to (<a href="" class="xref" data-knowl="./knowl/eq4_4.html" title="Equation 4.1.5">(4.1.5)</a>).</p></section></div></main>
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